Final answer:
After dilation centered at (3, 8) with a scale factor of 2, the line equation y = 3x - 1 remains unchanged because the center of dilation lies off the line and does not affect the slope or the y-intercept in this case. The correct answer is D. y = 3x - 1.
Step-by-step explanation:
The question involves understanding how a linear equation is transformed by a dilation centered at a specific point. The original line equation is given by y = 3x - 1. A dilation with a scale factor of 2, centered at the point (3, 8), does not change the slope of the line but it does affect the y-intercept. The center of dilation is not on the original line, which means the new line must go through this point.
To find the image of the line after the dilation, we note that the slope (m) remains the same since dilations centered at a point do not change slopes of lines. The new equation must satisfy the point (3, 8), so we use the slope-intercept form y = mx + b where m is the slope and b is the y-intercept.
The center of dilation gives us one point on the new line, so we use it to solve for the new y-intercept (b). Plugging in x = 3 and y = 8 gives us 8 = 3(3) + b, which simplifies to 8 = 9 + b. Solving for b gives us b = -1. This means the equation of the dilated line is y = 3x - 1, which is the same as the original equation due to the specific location of the dilation center.
Thus, the correct answer is D. y = 3x - 1, because the dilation about (3, 8) with a scale factor of 2 does not change the equation of the line.